large eddy simulation of combustion instabilities in turbulent

Center for Turbulence Research

Annual Research Briefs ��

��

Large eddy simulation of combustioninstabilities in turbulent premixed burners

By D� Veynante� AND T� Poinsot�

�� Motivations and objectives

Large Eddy Simulation �LES� techniques are viewed today as the next step inComputational Fluid Dynamics studies to address classes of problems where classi�cal Reynolds�averaged Navier Stokes approaches �RANS� have proved to lack pre�cision or where the intrinsically unsteady nature of the �ow makes RANS clearlyinadequate� In the �eld of combustion� the understanding and the control of com�bustion instabilities are domains where LES is required and will be applied in prac�tical systems� There are at least two reasons for this�

�� Reacting �ows submitted to instabilities are dominated by very large eddiessweeping the combustion chamber� Such �ows are obviously fully unsteady andmake RANS approaches dicult to use�

�� Structures controlling combustion in these �ows are large� and LES shouldbe easier in such cases than for turbulent combustion in general where an extendedrange of eddies has to be resolved to characterize the turbulence�chemistry inter�action�

Multiple techniques have been proposed in the past to perform LES of turbulentpremixed combustion �Menon and Kerstein �� Menon et al�� Smith andMenon �� Piana et al�� Veynante and Poinsot ��a�� Fewof them have been used in a realistic con�guration �see for example Kailasanathet al�� In most cases� fundamental studies in simple con�gurations such asfreely propagating �ames or stagnation point �ames have been performed� In suchsituations� assumptions are generally made in the fundamental studies �e�g� ignitionand quenching mechanisms are ignored� but must be reconsidered in more realisticcon�gurations� Flame stabilization and �ame�wall interactions� for example� shouldbe considered in detail and may in�uence the choice of the LES formulation�

We will brie�y recall the basis of LES techniques for combustion and investigatein more detail the performance of one speci�c method� the Thickened Flame ap�proach� proposed by Butler � O�Rourke �� Our objective is to test this methodin a con�guration where combustion instabilities occur� the �ame stabilized behinda backward�facing step� This con�guration was chosen since results from many ex�perimental studies are available �Keller et al�� Poinsot et al�� and since thecon�guration contains many features in common with real combustion chambers�Our attention will be also focused on �ame stabilization and �ame�wall interaction�

� Laboratoire EM�C� C�N�R�S� and Ecole Centrale Paris� France

� Institut de Mecanique des Fluides de Toulouse and CERFACS� France

�� D� Veynante � T� Poinsot

These issues have been addressed previously using DNS and RANS approaches�Poinsot �� Poinsot et al�� Bruneaux et al�� but have not received muchattention in the context of LES� Our �rst goal is to propose a LES technique whichcorrectly reproduces �ame wrinkling� at least when the �ow is dominated by largestructures� and handles �ame�wall interactions and stabilization regions in a phys�ical manner without ad hoc corrections�

The di�erent techniques proposed for LES of premixed combustion will be brie�ysummarized in Section � Our decision to investigate the thickened �ame �TF�model� initially proposed by O�Rourke and his coworkers �Butler � O�Rourke ��O�Rourke � Bracco �� will be explained�

Section � will present the con�guration studied and Section the numerical codeand the boundary conditions� The stabilization studies are described in Section�� and Section � presents �ame response to inlet velocity �uctuations� The e�ectof numerical parameters controlling this response �LES treatment� perturbationamplitude� thermal conditions on inlet sections� is also discussed�

�� LES techniques for turbulent premixed combustion

�� LES framework for combustion

Assuming that G is the LES �lter and x the location� any �ltered quantity Q isde�ned as�

Q �x� t� �

Z ��

��

Q �x� t�G �x � x�� dx� ��

For reacting �ows� a Favre �ltering is de�ned as�

� eQ � �Q �

Z ��

��

�Q �x� t�G �x � x�� dx� ��

where � is the �ltered density� The previous de�nition is similar to Favre averaging�widely used in RANS context�

Filtering the conservation equations controlling reacting �ows introduces un�known quantities to be modeled� �� guiuj � euieuj� the unresolved Reynolds stresses�

which requires a subgrid scale turbulence model� �� guiYk � euieYk� the unresolvedspecies �uxes� where a simple gradient expression is usually assumed�

guiYk � euieYk � ��TSc

� eYk�xi

��

with �T the subgrid kinematic turbulent viscosity and Sc the turbulent Schmidtnumber� �� guiT � eui eT � the unresolved heat �uxes� also modeled by a gradient ex�

pression� � � gYkT � eYk eT and gYkTn � eYk eTn� the species�temperature correlations�occurring when speci�c heats Cp are expressed in terms of polynomial approxima�tions of T � which are usually neglected� and �� the �ltered reaction rate ��k�

In the following� our attention will be focused on modeling the �ltered reactionrate ��k� The other terms have been addressed in previous studies� The Reynolds

LES of combustion instabilities ��

stresses are generally described using Smagorinsky or Germano dynamic modelswhereas unresolved turbulent transports are expressed with gradient expressions�No attempt has been yet conducted to take into account counter�gradient transportevidenced by theory �Libby � Bray �� Bray et al�� or DNS �Veynante etal�� Veynante � Poinsot ��b� in LES�

One diculty is encountered for large eddy simulations of premixed �ames� the�ame thickness ��l is in the range of approximately �� to �� mm and is generallysmaller than the LES mesh size �� Accordingly� species mass fraction and temper�ature pro�les are very sti� variables� thus the �ame front cannot be resolved on thecomputational mesh� To overcome this diculty� two main approaches have beenproposed� simulation of an arti�cially thickened �ame �TF� or use of a �ame fronttracking technique �G�equation��

�� Arrhenius law based on �ltered quantities �Arrhenius model�

A �rst simple model is to neglect subgrid scale contributions and to write thereaction rate as an Arrhenius law for �ltered quantities�

��F � A�� eYF eYO eT b exp

��TaeT�

� �

Such simple expressions assume perfect mixing at subgrid scales and implicitlyassume that turbulent time scales� �t� are shorter than chemical time scales� �c��t � �c�� The reaction zone thickness is also assumed suciently large to be re�solved on the LES mesh size� This formulation is generally used for reacting �owsin atmospheric boundary layers �Nieuwstadt� �� but is not relevant in mostcombustion applications� Segregation factors may be also introduced to correct Ar�rhenius expression to account for unmixedness� Speci�c Arrhenius�type expressionsincorporating combustion delays and changes due to subgrid scale mixing may alsobe derived in an ad�hoc manner� for example Kailasanath ��

�� The �eld equation �G model�

In this approach �Kerstein et al�� the �ame surface is described as anin�nitely thin propagating surface �i�e� �amelet�� In using this approach� onetracks the position of the �ame front using a �eld variable G� The �ame surfaceis associated with a speci�c isolevel G � G�� The gradients in the G��eld can bemuch smoother than those of the progress variable c to the point where they can beresolved on the LES mesh� Work in progress on the use of G equation has shown thepotential but also the diculties of this approach �see Bourlioux et al�� Pianaet al�� Im et al�� Veynante and Poinsot ��a��

�� Random vortex methods �RVM model�

Random vortex methods are another class of models suitable for LES of premixedcombustion� In this grid�free approach� chemistry may be handled in a Lagrangianmanner by following �ame elements� Examples of such approaches may be foundin Ghoniem et al��


�� The thickened ame model �TF model�

The key idea of the thickened �ame �TF� model is to consider a �ame havingthe same laminar �ame speed sl but a larger �ame thickness than the actual actual�ame in order to be resolved on the LES computational grid �Butler and O�Rourke�� O�Rourke and Bracco �� Following simple theories of laminar premixed�ame �Williams �� Kuo �� the �ame speed sl and the �ame thickness �l maybe expressed as�

sl �pa �W � �l �

a

sl

where a is the thermal di�usivity and �W the total reaction rate� Then� an increaseof the �ame thickness ��l by a factor F while maintaining a constant �ame speeds�l may be achieved by replacing the thermal di�usivity a by Fa and the reaction

rate �W by �W�F as summarized in Table I� Numerically� such a transformation isperformed simply by dividing the pre�exponential constant and the Prandtl and theSchmidt numbers by the thickening factor F �

Table I� Comparison between normal �superscript �� and thickened �ame �super�script �� The thickening factor is F �

Flame Flame Preexponential Prandtl Schmidtspeed thickness Factor

Normal �ame s�l ��l A� P �r S�c

Thickened �ame s�l � s�l ��l � F��l A� � A��F P �r �F S�c�F

For suciently large values of the factor F � the thickened �ame front may beresolved on the LES computational mesh� In practical applications� values of ��l�estimated by ��l s

�L�a � � are of the order of �� to � mm so that thickening factors

F of the order of � to �� should suce for many practical simulations� Based onArrhenius law� the TF model has the advantages that it can handle ignition and�ame�wall interaction processes without any sub�model�

However� thickening the �ame front may have the following two undesired e�ects�First� the �ame propagation may be a�ected when small scales are present in the�ow because these structures could become unable to wrinkle the thickened �amefront �Poinsot et al�� For combustion instabilities� this drawback may not becrucial because of the large values of the ratio of the vortex size L to the �amethickness ��l � Second� the sensitivity of the �ame to stretch is also increased by Fbecause of the transformation� The thickened �ame will react to a stretch of �F asthe actual �ame would to a stretch of � Many DNS of turbulent premixed combus�tion have suggested that� in the mean� the e�ect of stretch on the local �amelets wasnot strong �Haworth � Poinsot �� Baum et al�� Trouv�e � Poinsot �� butincreasing this e�ect by a factor F of the order of �� may have unexpected e�ects�


for example on quenching� Note however that this diculty is also encountered inthe G�equation approach where strain e�ects on the displacement speed have to beintroduced in an ad�hoc fashion because the G model is� by construction� insensitiveto stretch�

�� Objectives and con�guration

Our objective in the present work is to investigate the limits of the TF model forlarge eddy simulations of combustion instabilities in premixed burners� The issuesmentioned before will be analyzed by computing the same �ow �prototype of acombustion instability in a premixed burner� with both �normal� and a �thickened��ame descriptions� The normal �ame is described using a classical DNS formulation�For thickened �ame simulations� no LES model is used for the �ow itself� ourobjective is just to qualify the TF approach independently of the LES turbulencemodel� Furthermore� for the present two�dimensional simulations� no small�scaleturbulence is present� These TF simulations are in fact DNS where the �amecharacteristics have been changed according to the relations summarized in Table I�For future studies in three�dimensional �ows� an additional coupling model betweenthe turbulence LES model and the TF model should be incorporated�

INLET

SYMMETRY AXIS

SYMMETRY AXIS

OUTLET

x=0 x=Xmaxy=0

y=b

y=l

WALL

Figure �� Numerical con�guration corresponding to the premixed propane�airburner used by Poinsot et al�� to investigate combustion instabilities� Theactual burner has �ve injection slots�

The numerical con�guration� displayed in Fig� �� corresponds to the turbulentpremixed propane�air experimental burner of Poinsot et al�� This burneris dominated by multiple instability modes corresponding to acoustic eigenmodesof the whole combustion system �including compressor and inlet pipes�� Observedfrequencies range between � Hz and �� Hz� The strongest mode occurs at �� Hzfor an equivalence ratio of �� and a total �ow rate �m of �� g�s ��ow conditionsare summarized in Table II��


Table II� Flow conditions for the �� Hz instability mode �Poinsot et al��U� is the inlet velocity� c� the sound speed� �� is the kinematic viscosity of freshgases� l and b are respectively the burner and the step thickness �see Fig� ��

s�l ��l U� c� Re � c�l�� U��l � b�� Frequency�m�s� �m� �m�s� �m�s� acoustic inlet �Hz�

��

For the �ow conditions in Table II� the whole system resonates at a frequencyof �� Hz while mushroom�like vortices are shed at the same frequency from all�ve injection slots� These structures are not created by hydrodynamic instabilities�which are also observed but at a higher frequency� but are due to strong simulta�neous velocity surges in the �ve injection slots� These vortices grow� are convected�and interact with vortices issued from neighboring slots� leading to small�scale tur�bulence and intense heat release� The time delay between the velocity surge leadingto the formation of these vortices and the peak heat release is an essential param�eter for all combustion instability models �see Crocco � Cheng �� Crocco ��McManus et al�� Candel et al�� Estimating this delay does not require oneto take into account the whole system and the acoustics which induce the vortexformation itself� A proper strategy is to create a vortex by pulsating the combus�tion chamber inlet �ow �eld and to study the e�ect of this vortex on the overallcombustion process�

�� Numerical technique and protocol to study ame response

This study is conducted using the NTMIX code� a two�dimensional DNS solverdeveloped by CTR and Ecole Centrale Paris and described in Veynante � Poinsot�� or Veynante et al�� The full compressible reacting Navier�Stokes equa�tions are solved assuming perfect gases with constant molar mass and a speci�cheat ratio � � �� The thermal conductivity � and the di�usion coecient D areobtained from the dynamic viscosity coecient according to

� � Cp�Pr and D � ��Sc��

where the Prandtl number Pr and the Schmidt number Sc are constant� As aconsequence the Lewis number Le � Sc�Pr is also constant� The viscosity is afunction of temperature according to � u�T�Tu�n where n � ��

The computational domain is Lx � Ly with Nx � Ny grid points� Boundaryconditions �Fig� �� correspond to a partially blocked inlet �blockage ratio b�l of�� on the left� non�re�ecting boundary conditions on the right� and symmetricboundaries on both sides� The velocity pro�le at the inlet is a tanh function witha thickness �Y �

An important diculty in studying combustion delays is that excitation proce�dures have to be performed on a given baseline �ow� Either this baseline �ow is


stable and the �ame response might not correspond to the one expected for unstablecases� or the baseline �ow is unstable and measuring a transfer function becomesextremely dicult because the �ow is dominated by its own instability �similardiculties are encountered for DNS of non�reacting �ows in absolutely unstableregimes�� Experimentally� the only possible approach is the �rst solution� Poinsotet al�� for example� have measured the re�ection coecient of a premixed�ame but only for stable regimes close to instability� Numerically� however� it ispossible in certain cases to create a stable baseline �ow in a regime which shouldbe unstable by slightly changing the expression of the reaction rate� Indeed� theo�retical studies of combustion instabilities indicate that one main factor promotinginstabilities is the dependence of the reaction rate on pressure �i�e� on density� asevidenced by the Rayleigh criterion which states that instability occurs when pres�sure and total heat release oscillate in phase �Crocco �� McManus et al��Poinsot and Candel �� for example� veri�ed numerically that an anchored �amewas more likely to become unstable when �ame speeds were pressure dependent�This suggests the following approach� Assuming that fuel is the de�cient speciescontrolling the reaction rate ��F �

��F � B�YF exp

��TaT

��

then for the pressure sensitive �PS� case� any pressure increase �corresponding toan increase of the density �� will also increase the reaction rate� To inhibit thise�ect� a second expression called PI �pressure insensitive� is also used�

��F � BP�rT

YF exp

��TaT

��

where P� is a constant reference pressure� The PI expression makes the reactionrate insensitive to pressure waves and cuts an important link in the combustioninstability loop�

The PS and PI expression will give the same results for a stable �ame �for exam�ple� an uncon�ned �ame�� However� only the PI expression can produce a stablebaseline �ow in ducted �ows�

�� Stabilization and baseline ows for ducted ames

The di�erent runs presented in this report are summarized in Table III� Runs B�S � S� and S� correspond to DNS ��ames with normal thickness ��l � while runs M��S�� S�� and S� correspond to thickened �ames �by a factor F ranging from �� to�� Runs B� M�� S�� and S� correspond to the same physical �ow �Re � ��where the computation is performed with DNS for B �F � �� and with variousvalues of the thickening factor F � �� M�� S�� and �� S�� Runs S � S�� andS� correspond to DNS with a lower Reynolds number �Re � �� Two thermalconditions have been tested for the blockage wall lying between y � � and y � b atthe inlet �Fig� �� This wall may be adiabatic �adiabatic wall� called AW� or cooled


with an imposed temperature Tw � T� where T� is the inlet gases temperature�cooled wall� called CW in Table III��

Table III� Flow conditions for the DNS and LES of ducted �ames� For all �ows�U��c� � �� the Reynolds number Re is � c�l�� the temperature change throughthe �ame front is T��T� � �� T� � T��T� � �� the sound speed in thefresh gas is c� �and c� in the burnt gas�� the activation temperature Ta is suchthat � � �Ta�T� � �� the box height is l�L � �� the �ame Mach number s�l �c� is�� the blockage is �� b�l � �� l is the �ame thickness after thickening��l � F ��l ��

RUN l��l Re F RR Wall �Y �l Xmax�l Nx Ny

cond� cond�

S� �� PS AW �� S �� PI CW �� S� �� PI CW �� S� �� PI AW �� S� �� PI AW �� S� �� PI AW �� M� �� PI AW �� B � �� PI AW ��

These conditions do not correspond exactly to the experiment of Poinsot etal�Although the geometry is the same� the Reynolds number of the largest sim�ulation �B� is only one third of the experiment� Our goal� however� is to validatethe TF methodology and� for the moment� no detailed comparisons are performedwith the experiment�

�� E�ects of formulations of �� on stabilization

All computations are initialized with an oblique �ame starting behind the step us�ing temperature and fuel mass fraction pro�les corresponding to a one�dimensionallaminar premixed �ame for the same equivalence ratio� Starting from this �eld�the simulation evolves without external excitation until a steady state is reachedor when a well established oscillation is found� Figure compares the temporalevolution of the total reaction rate for PS and PI formulations in cases S� and S��The PS formulation leads to oscillations both in transverse �fl�c� � �� and longi�tudinal modes �fl�c� � �� of the computational box� On the other hand� the PIformulation leads to a constant reaction rate and a steady �ame regime�

�� E�ect of inlet wall thermal condition

Figure � shows velocity vectors and the reaction rate �eld for run S� in the vicinityof the injection slots� This �ow exhibits a recirculation zone having a length of about��b� The gas temperature inside this recirculation zone controls the anchoring of


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2 01 51 050

Reduced time (Uo t/l)

( a )

( b )

Figure �� E�ect of reaction rate formulation on �ow stabilization� �a� case S��pressure insensitive form� �b� case S�� pressure sensitive form� Time evolution oftotal burning rate �adiabatic walls��

Figure �� Zoom on velocity vectors and reaction rate �eld in the vicinity of theinjection slot �Cold Wall � run S ��


(a)

(b)

Figure �� Reaction rate �eld� E�ect of wall condition on the �ame stabilization��a� Cold wall �CW� run S �� b� Adiabatic wall �AW� run S��

the �ame� These hot gases are produced by combustion and recirculated behindthe step but may also be cooled by the wall� Therefore the thermal conditions onthe wall between y � � and y � b are important parameters �see Fig� ��

Any heat losses in the vicinity of the recirculation gas have a strong e�ect on thesteady �ame position but also on its response to unsteady pulsations� Two thermalconditions have been tested for this wall� The adiabatic wall �AW� condition �runS�� allows the �ame to start on the wall while the cooled wall �CW� condition�run S � inhibits reaction near the wall and forces the �ame to be lifted as shownby the reaction rate �elds in Fig� � The �ame length is also increased and thecharacteristic �ame time is changed �see next Section��

�� Transfer function of premixed ducted ames

�� Methodology

Once stable �ames are obtained� their transfer function may be studied by inject�ing acoustic disturbances through the inlet� The inlet velocity pro�le is modulatedhere according to the following expression�

U�x� y� t� � Usteady

�� U�

inc exp

��

�t � ttrigtwidth

��

��

Examples of a �ame excited with U�

inc � �� U�ttrig�l � �� and U�twidth�l � ��are displayed in Fig� � for run S� �see table III�� The formation of a large reactingvortex is observed� The shape of this vortex is similar to the mushroom vorticesobserved in the experiment of Poinsot et al�� Because of the AW conditionused for the wall� the �ame remains anchored at all times on the wall�

Figure � shows that the total heat release lags the inlet �ow rate by a delay� � ��l�U�� This delay directly controls the instability modes since the period Tof most combustion instabilities is of the order of � �Crocco � Cheng �� Poinsotet al�� However� the delay obtained through such a simulation must be usedwith caution because it depends on multiple parameters� In the following� threeparameters will be investigated� �� the thickening parameter F of the TF model�


t=4

t=8

t=12

t=16

(a)

(b)

(c)

Figure �� Pulsated premixed �ame �S�� computed using the TF model �thick�ening factor F � �� Temperature �eld� �a� U�t�l � �� b� U�t�l � �� c�U�t�l � � �

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6Normalized time variations

543210


Figure �� Time evolution of inlet �ow rate � �� total reaction rate � �and outlet �ow rate � �� Run S� with F � ��


corresponding to the LES treatment of the reaction rate� �� the wall condition�cold or adiabatic�� and �� the excitation amplitude U�

inc�

�� E�ects of LES treatment �thickening factor F �

The Re � �� case was computed with thickening factors F � � �run B� �� grid �� points�� F � �� M�� points�� F � � �S�� points�and F � �� S�� points�� As the �ame becomes larger �F is increased��the grid size Nx � Ny may be reduced� decreasing the computational time� Theexcitation parameters are U�

inc � �� U�ttrig�l � �� and U�twidth�l � �� Figure �shows the time evolution of the total reaction rate for these four simulations�

1.8

1.6

1.4

1.2

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0.8

76543210

Reduced time (Uot/l)

Figure �� E�ects of the thickening factor F �LES treatment of the reaction rate��The normalized total burning rate is displayed versus time� Comparison betweenDNS� F � � �run B�� LES with F � �� M�� F � � �S�� and F � �� S�� Excitation parameters are U�

inc � �� U�ttrig�l � �� andU�twidth�l � ��

The general evolution of all �ows is similar� di�erences of the order of �� areobserved for the total reaction rate� However the shapes of the reaction rate curvesare di�erent� the DNS �F � �� burns initially faster than the LES cases but moreslowly in the late stages of the interaction� This �nding is consistent with the TFformalism� less �ame surface is created for LES runs than for DNS� leading to areduced combustion� because the sensitivity of a thickened �ame to a hydrodynamicperturbation is lower than the one of a thin �ame as shown by Poinsot et al��Later on� however� the additive reactants injected during the excitation have toburn� leading to a larger combustion rate in the LES computations�

Figures � and � display �elds of instantaneous reaction rates for the four compu�tations at times U�t�l � �� and U�t�l � �� respectively� These plots con�rm thethickening of the reaction zone when the factor F is increased� This thickening af�fects �ame wrinkling in complex ways� the DNS creates a �rst pocket of fresh gases


in the burnt products earlier than the three LES runs� More �ame surface is alsogenerated� This �ame surface is mainly due to small scale wrinkling� which leadsto the formation of small pockets� These pockets are burned out rapidly� On theother hand� less small pockets are initially created in the LES cases so that a largepocket of fresh reactants is consumed later� This phenomenon becomes dramaticfor the F � �� LES run where the evolution of the �ow after U�t�l � �� di�ersfrom the DNS result by �� because most of the �ame wrinkling is missed�

(a)

(b)

(c)

(d)

Figure �� Instantaneous reaction rate �elds at time U�t�l � �� Comparisonbetween DNS �a�� LES with F � �� b�� LES with F � � �c� and LES with F � ��d�� See caption of Fig� � for runs characteristics�

�� E�ects of inlet wall condition

Figure �� compares two DNS simulations �runs S� and S�� for an excitationcorresponding to U�

inc � �� U�ttrig�l � �� and U�twidth�l � �� Run S� is similar toS �only the box length is di�erent� and performed with a cold wall �CW� conditionon the inlet wall while S� assumes an adiabatic wall �Reaction rate contours for both�ames under steady conditions are displayed on Fig� ��

Figures �� and � display �elds of reaction rates for these two computations atreduced times U�t�l � � � �� and �� The �ame stabilized behind a cold


(a)

(b)

(c)

(d)

Figure � Instantaneous reaction rate �elds at time U�t�l � �� Comparisonbetween DNS �a�� LES with F � �� b�� LES with F � � �c� and LES with F � ��d�� See caption of Fig� � for runs characteristics�

wall �Fig� �� reacts later to excitation than the adiabatic �ame �Fig� �� but morestrongly� Large di�erences both in delay and amplitude are observed� demonstratingthe importance of the condition chosen for the inlet wall� Two factors explain thesedi�erences� �� the CW �ame lies closer to the recirculating burnt region than theAW �ame and� therefore� feels� the vortex with less amplitude and at a later timethan the AW �ame� and �� the reaction rate of the CW �ame is very small nearthe injection slot because of heat losses while the AW �ame burns everywhere withthe laminar �ame speed� The resulting pattern of this �ame�vortex� interactionis� therefore� extremely di�erent� The e�ects of this boundary condition appear tobe as strong as the thickening factor F used in the TF model� This shows thatthe LES of the purely propagating �ame �handled with the TF model� is only oneaspect of CFD for combustion instabilities of con�ned �ames and that factors suchas boundary conditions and �ame stabilization could play a crucial role�

LES with cold wall conditions are not presented here because speci�c treatmentsof wall heat �uxes will be required for these cases� Dividing Pr by a factor Fthickens the �ame but also increases heat transfer to the walls by the same factor


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8 06 04 02 00


Figure �� E�ects on inlet wall condition� Normalized total burning rate versustime� Comparison between adiabatic wall �AW � S�� and cold wall �CW �S��

(a)

(b)

(d)

(c)

Figure �� Fields of reaction rate for simulation S� �DNS� adiabatic wall�� a�U�t�l � � � �b� U�t�l � �� c� U�t�l � �� and �d� U�t�l � ��


(a)

(b)

(d)

(c)

Figure �� Fields of reaction rate for simulation S� �DNS� cold wall�� a�U�t�l � � � �b� U�t�l � �� c� U�t�l � �� and �d� U�t�l � ��

F � This could be avoided by an adequate treatment of wall �uxes but remains tobe investigated�

�� E�ects of excitation amplitude

Most models for combustion instability are linear� It is� however� well�knownthat �ame response is strongly non�linear� and the excitation amplitude to pulsate�ames has to be chosen carefully� Figure �� shows �ame response for an LESrun �S�� F � �� for three excitation levels� U�

inc � �� and �� In all cases�U�ttrig�l � �� and U�twidth�l � �� For low levels of excitation amplitude �� of

incoming velocity for U�

inc � �� the delay is much longer than it is in cases withmore intense perturbations� Examination of the instantaneous �ow �elds revealsthat no pocket is formed for case U�

inc � �� Flame wrinkling is higher in the twoother cases where the �ame is shred by the vortical �eld�

�� Theoretical analysis of the thickened ame response

The response of a thickened �ame to an excitation is now analyzed� Our objectiveis to propose a simple model explaining the �ndings of Fig� � and �� Under the�amelet assumption� the �ame stretch K measures the increase in �ame surfacearea A �Candel � Poinsot� ��

K ��

A

dA

d t��


1.8

1.6

1.4

1.2

1.0

0.8

86420

Reduced time (Uo t /l)

Figure �� E�ects of excitation amplitude� Normalized total burning rate versustime for U�

inc � �� U�

inc � �� and U�

inc � � � �� Run S��F � ��

2.0

1.8

1.6

1.4

1.2

1.0

0.8

43210

Reduced time (Uot/l)

Figure �� Analysis of the initial �ame response� The normalized total burningrate is displayed versus time� Bold lines correspond to numerical results �alreadydisplayed on Fig� �� thin lines � � are �ts according to Eq� �� DNS� F � ��run B�� LES with F � �� M�� F � � �S�� and F � ��S�� Excitation parameters are U�

inc � �� U�ttrig�l � �� and U�twidth�l ��


0.8

0.6

0.4

0.2

0.03 4 5 6 7 8 9

12 3 4 5 6 7 8 9

1 02 3 4 5 6 7 8 9

1 0 0

Figure �� Estimation of the eciency function C �see Eq� �� plotted as afunction of the length scale ratio r��l between the size of the perturbation and thearti�cial thickness of the �ame� The continuous line � � corresponds to theeciency function CMP estimated by Meneveau and Poinsot �� Excitationparameters are U�ttrig�l � �� U�twidth�l � �� and U�

inc � �� U�

inc � �� and U�

inc � ��

Assuming that the local reaction rate per unit of �ame surface area is almost con�stant along the �ame front� the total reaction rate �W is proportional to the �amearea A and depends on the mean stretch in the same way�

In our simulations� the mean stretch is mainly due to the strain � which is inturn due to the vortex generated by the inlet velocity perturbation� But� followingMeneveau and Poinsot �� an eciency function C depending on the ratiobetween the vortex size r and the �ame thickness �l has to be introduced to takeinto account the reduced ability of small vortices to wrinkle the �ame front� Then�integrating Eq� �� leads to an estimate of the time evolution of the reduced totalreaction rate �W� �W�� where �W� corresponds to the total reaction rate under steadystate operation�

�W�W�

� exp

�C

�r

�l

� t

��

Estimates of the vortex size r and the strain rate are now required� One maypropose�

r � U� twidth � � U�

inc

U�r

��

The eciency function C�r��l� may now be estimated by �tting the expression�� on reduced reaction rates displayed on Fig� � and �� Reaction rate �ts aredisplayed against numerical simulations on Fig� � �Only the growing phase of thetotal reaction rate is used here�� The agreement is quite satisfactory� On Fig� ��the eciency function C�r��l� is plotted as a function of the length scale ratio r��land compared to the eciency function CMP proposed by Meneveau and Poinsot


�� and extracted from the �ame�vortex interaction DNS conducted by Poinsot etal�� As expected� the eciency function C is found to decrease with the lengthscale ratio r��l with a shape similar to CMP � The discrepancy between C and CMP

is probably due to the di�erence between present simulations and those of Poinsot etal��two vortices interacting with a normal �ame in Poinsot et al�DNS� one velocityperturbation interacting with an oblique �ame here�� leading to di�erences in theestimation of r and � Another point is that� in our simulation� the �ame structureis modi�ed by decreasing the pre�exponential factor B and the Schmidt numberSc� keeping the hydrodynamic perturbation constant� In the simulations of Poinsotet al�� the �ame structure remains unchanged whereas vortex size and strengthare modi�ed� Nevertheless� this �nding is very interesting because the eciencyfunction C�r��l �� and more generally an ITNFS�like formulation �see Meneveauand Poinsot� �� could be implemented in the reaction rate expression to correctthe reduced ability of a thickened �ame to be wrinkled by small structures� Thispoint could be investigated from DNS of �ame�vortex interactions using variousvalues of the thickening factor F �

�� Conclusion

The forced response of a �ame stabilized behind a step in a geometry correspond�ing to the experiment of Poinsot et al�� has been studied at di�erent Reynoldsnumbers using Direct Numerical Simulations and Large Eddy Simulations basedon the Thickened Flame model� This model allows the computation of a premixed�ame on a coarse grid by increasing its thickness while maintaining its �ame speed�The TF model is able to compute the �ame response within �� when thickeningfactors as large as � are used� Even though the TF model modi�es the �ame re�sponse� its in�uence is smaller than other parameters which are speci�cally linkedto combustion instabilities in dump�stabilized �ames� the excitation amplitude andthe thermal condition on the wall near the injection slot� for example� are found tohave comparable e�ects on the �ame response� An eciency function� similar to theITNFS formulation proposed by Meneveau and Poinsot �� could be introducedto reduce the modi�cation of the �ame response due to the TF model� Accordingly�the thickened �ame �TF� approach seems to be a good compromise for large eddysimulations of combustion instability in premixed burners�

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